In problems 1-6, determine the convergence set of the given power series.
The set is,
Use the ratio test to determine the radius of convergence (with ):
The radius of convergence is 1, therefore convergent set for the given power series is.
To completely identify the convergence set, we have to check whether the boundary points -1 and -3 are included in the set or not.
Checking at , by substituting the value ofx by -1 .
The above series is divergent, thus the point -1 is excluded from the convergent set
Similarly, checking at by substituting the value of x by -3.
The above series is divergent, thus the point -3 is not included in the convergent set. The convergent set for the given power series is.
Show that fn(0)=0 for n=0,1,2.... and hence that the Maclaurin series for f(x) is 0+0+0+.... , which converges for all x but is equal to f(x) only when x=0 . This is an example of a function possessing derivatives of all orders (at x0 =0 ), whose Taylor series converges, but the Taylor series (about x0 =0) does not converge to the original function! Consequently, this function is not analytic at x=0.
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